Homework 6 Due: Oct 15 These first two should probably have been assigned earlier. 1. If X is a topological space, the topological dimension of X, tdim(X), is the supremum of the lengths of all chains Z0 ( Z1 ( ··· ( Zn, where each Zi is a closed, irreducible subset of X. If R is a ring, the height of a prime ideal p ⊂ R is the supremum of the lengths of all chains p0 ( p1 ( ··· pn = p of distinct prime ideals. The Krull dimension of R, kdim R, is the supremum of the heights of all prime ideals. Let Y be an irreducible affine variety. (a) Prove that tdim(Y) = kdim(k[Y]). (b) Use the following result from commutative algebra to show that tdim(Y) = dim(Y). Theorem Let R be an integral domain which is finitely generated as a k-algebra. Then kdim R = tr. deg.(Frac(R)/k). 2. Let V and W be k-vector spaces of dimensions m and n, respectively. After choosing a basis on V and W, we may identify Amn with (the set of m × n matrices with entries in k, and thus with) LinMap(V, W). (a) Suppose dim V = dim W. Prove that the set of elements of LinMap(V, W) which are actually isomorphisms is a Zariski open subset of LinMap(V, W). (b) Let r be a nonnegative integer. Show that the set Mr := {α ∈ LinMap(V, W) : dim(α(V)) ≤ r} is a Zariski closed subset of LinMap(V, W). (Here, dim means dimension as vector space.) 3. A ring variety is a variety X equipped with addition A morphism α : X × X → X; multiplication A morphism µ : X × X → X; additive inverse A morphism ι : X → X; additive identity An element ζ ∈ X Professor Jeff Achter M672: Algebraic geometry Colorado State University Fall 2008 such that (X,α,ζ) is a group variety, and multiplication satisfies the obvious axioms. Suppose that X is a projective irreducible ring variety. Prove that the multiplication map must be trivial. You may, and should, use basic properties of rings. 4. Elimination theory, revisited Suppose Z = ZPm×An (I), where I ⊂ k[X0, ··· , Xm, y1, ··· , yn] is homogeneous in the Xi’s. The projective elimination ideal is d Iˆ = {h ∈ k[y1, ··· , yn] : ∃d > 0 : (X0, ··· , Xn) h ⊂ I}. Suppose [a0, ··· , am] × (b1, ··· , bm) ∈ Z and h ∈ Iˆ. Show that h(b1, ··· , bm) = 0. ˆ ˆ This shows p2(Z) ⊆ ZAn (I). In fact, one can show that p2(Z) = ZAn (I), which provides a proof m n n that the map p2 : P × A → A is closed. 5. Consider the morphism φ P1 - P2 [a, b] - [a3, a2b + ab2, b3] This induces a map of homogeneous coordinate rings 2 - 1 kh[P ] kh[P ] X - S3 Y - S2T + ST2 Z - T3 (a) Describe the bihomogeneous ideal I ⊂ k[S, T, X, Y, Z] such that Z(I) = Γφ, the graph of φ. 1 I 2 (φ( )) (b) Explain, in the notation of the previous problem, how to compute P P , the ideal which defines the image of φ. 1 I 2 (φ( )) Extra: Using a computer algebra package if you like, compute P P . Professor Jeff Achter M672: Algebraic geometry Colorado State University Fall 2008.